If you want to get out ASAP, once you recieve your orders, elect to sell your leave balance instead of going on terminal. You can also opt for 0 days for PTDY that's given from your commander. I turned down 30 days of PTDY so that I could get out of the military sooner and start my VA effective date in June. However, you could also just take any terminal and PTDY if you believe that your command would leave you alone for that entire time.
As for the exam things, I would start to track down: your msc, the exam provider, and talk with the doctor that actually did the exam. Be a strong advocate for yourself. These people don't care about your case or life as much as you do. My case itself had many issues that I had to resolve myself and figure out. Right now, it's a little too late to try and get your exams re-done since you're waiting for the DOD rating. However, you can file an HLR once you get out. But use that NAR that you will get and see exactly what the VA tells you. They will give you word for word why you're denied or got a specific rating. Use that to your advantage. The best thing is that you would get back pay on any conditions that are increased.
As for the sleep study, the VA usually require one. I had one done at a DOD facility and the VA / VES almost made me go to complete another one. Thankfully they just accepted the one that I already did.
And no worries! I know how stressful this process is. I've been anxious to figure out my VA rating, then the DOD rating, and now I'm anxious to know when my orders will drop.